There has been some talk about this in the exercise world over the last few years, as there has been a push by supplement manufacturers to convince people that it is a needed item. The theory is that since it transports fatty acids to be broken down, more carnitine will transport more, thus driving increased fat utilization/breakdown.
L-Carnitine is a protein that is manufactured in the body, from the amino acids lysine and methionine by the kidneys and the liver. It can also be consumed, and it is mostly stored in tissues such as skeletal and cardiac muscle. It plays a role in delivering long chain fatty acids to mitochondria for energy metabolism, as well as transporting out waste products. The body excretes excess carnitine to maintain stable blood levels, so unless you are deficient (Or have a condition where your body does not independently produce it), you are unlikely to benefit from extra carnitine consumption.
I was able to find two different studies regarding the effect on weight loss. The first study was performed using obese rats fed a low calorie (hypocaloric) diet. One group was not given a supplement, the other group had added carnitine. At the end of the study, while there was weight loss and favorable fat to protein ratio shifts for both groups, the carnitine group did not produce a greater loss of weight.
The second study was similar, except it was performed using moderately overweight women, divided into two groups. One group consumed a placebo, the other carnitine, and they all did the same exercise program. In the end, there was no statistical difference between the groups for weight loss and body composition. In addition, 5 of the 18 (28%) in the carnitine group had to drop out due to diarrhea and nausea.